Olivier Delestre

SWASHES: a compilation of shallow water analytic solutions for hydraulic and environmental studies

Numerous codes are being developed to solve Shallow Water equations. Because there are used in hydraulic and environmental studies, their capability to simulate properly flow dynamics is critical to guarantee infrastructure and human safety. While validating these codes is an important issue, code validations are currently restricted because analytic solutions to the Shallow Water equations are rare and have been published on an individual basis over a period of more than five decades. This article aims at making analytic solutions to the Shallow Water equations easily available to code developers and users. It compiles a significant number of analytic solutions to the Shallow Water equations that are currently scattered through the literature of various scientific disciplines. The analytic solutions are described in a unified formalism to make a consistent set of test cases. These analytic solutions encompass a wide variety of flow conditions (supercritical, subcritical, shock, etc.), in 1 or 2 space dimensions, with or without rain and soil friction, for transitory flow or steady state. The corresponding source codes are made available to the community (http://www.univ-orleans.fr/mapmo/soft/SWASHES), so that users of Shallow Water-based models can easily find an adaptable benchmark library to validate their numerical methods.

A ‘well-balanced’ finite volume scheme for blood flow simulation

We are interested in simulating blood flow in arteries with a one dimensional model. Thanks to recent developments in the analysis of hyperbolic system of conservation laws (in the Saint-Venant/ shallow water equations context) we will perform a simple finite volume scheme. We focus on conservation properties of this scheme which were not previously considered. To emphasize the necessity of this scheme, we present how a too simple numerical scheme may induce spurious flows when the basic static shape of the radius changes. On contrary, the proposed scheme is well-balanced: it preserves equilibria of Q = 0. Then examples of analytical or linearized solutions with and without viscous damping are presented to validate the calculations. The influence of abrupt change of basic radius is emphasized in the case of an aneurism.

A Numerical Scheme for a Viscous Shallow Water Model with Friction

We consider a particular viscous shallow water model with topography and friction laws, formally derived by asymptotic expansion from the three-dimensional free surface Navier-Stokes equations. Emphasize is put on the numerical study: the viscous system is regarded as an hyperbolic system with source terms and discretized using a second order finite volume method. New steady states solutions for open channel flows are introduced for the whole model with viscous and friction terms. The proposed numerical scheme is validated against these new benchmarks.

Overland Flow Modeling with the Shallow Water Equations Using a Well-Balanced Numerical Scheme: Better Predictions or Just More Complexity

In the last decades, several physically based hydrological modeling approaches of various complexities have been developed that solve shallow water equations or their approximations using various numerical methods. Users of the model may not necessarily know the different hypotheses underlying these development and simplifications, and it might therefore be difficult to judge if a code is well adapted to their objectives and test case configurations. This paper aims to compare the predictive abilities of different models and evaluate potential gain by using an advanced numerical scheme for modeling runoff. Four different codes are presented, each based on either shallow water or kinematic wave equations, and using either the finite volume or finite difference method. These four numerical codes are compared with different test cases, allowing to emphasize their main strengths and weaknesses. Results show that, for relatively simple configurations, kinematic wave equations solved with the finite volume method represent an interesting option. Nevertheless, as it appears to be limited in case of discontinuous topography or strong spatial heterogeneities, for these cases they advise the use of shallow water equations solved with the finite volume method.

Well-balanced finite difference weighted essentially non-oscillatory schemes for the blood flow model: Well-balanced finite difference weighted essentially non-oscillatory schemes for the blood flow model

The blood flow model maintains the steady state solutions, in which the flux gradients are non-zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted non-oscillatory (WENO) schemes to this model with such well-balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well-balanced property, and keep good resolution for smooth and discontinuous solutions.

Well‐balanced finite difference WENO schemes for the blood flow model

The blood flow model maintains the steady state solutions, in which the flux gradients are non-zero but exactly balanced by the source term. In this paper, we design high order finite difference weighted non-oscillatory (WENO) schemes to this model with such well-balanced property and at the same time keeping genuine high order accuracy. Rigorous theoretical analysis as well as extensive numerical results all indicate that the resulting schemes verify high order accuracy, maintain the well-balanced property, and keep good resolution for smooth and discontinuous solutions.

FullSWOF: Full Shallow-Water equations for Overland Flow

Numerical simulations of shallow flows are required in numerous applications and are typically performed by solving shallow-water equations. FullSWOF solves these equations by using up-to-date finite volume methods and well-balanced schemes. Several features make FullSWOF particularly suitable for surface water hydrologists: small water depths and wet-dry transitions are robustly addressed, rainfall and infiltration are incorporated, and grid-based digital topographies can be used directly. The modular structure of FullSWOF is also useful to numerical modelers willing to test new schemes or boundary conditions.

Water planning in a mixed land use Mediterranean area: point-source abstraction and pollution scenarios by a numerical model of varying stream-aquifer regime

Integrated hydrodynamic modelling is an efficient approach for making semi-quantitative scenarios reliable enough for groundwater management, provided that the numerical simulations are from a validated model. The model set-up, however, involves many inputs due to the complexity of both the hydrological system and the land use. The case study of a Mediterranean alluvial unconfined aquifer in the lower Var valley (Southern France) is useful to test a method to estimate lacking data on water abstraction by small farms in urban context. With this estimation of the undocumented pumping volumes, and after calibration of the exchange parameters of the stream-aquifer system with the help of a river model, the groundwater flow model shows a high goodness of fit with the measured potentiometric levels. The consistency between simulated results and real behaviour of the system, with regard to the observed effects of lowering weirs and previously published hydrochemistry data, confirms reliability of the groundwater flow model. On the other hand, accuracy of the transport model output may be influenced by many parameters, many of which are not derived from field measurements. In this case study, for which river-aquifer feeding is the main control, the partition coefficient between direct recharge and runoff does not show a significant effect on the transport model output, and therefore, uncertainty of the hydrological terms such as evapotranspiration and runoff is not a first-rank issue to the pollution propagation. The simulation of pollution scenarios with the model returns expected pessimistic outputs, with regard to hazard management. The model is now ready to be used in a decision support system by the local water supply managers.

Numerical Scheme for a Viscous Shallow Water System Including New Friction Laws of Second Order: Validation and Application

In this work, we are interested in the derivation of a new shallow water model with a diffusion source term. Analytical solutions for steady flow regimes are first presented to validate a numerical method designed to solve this new model. Then this model is applied on real data and seems to give better results than the classical shallow water system.

A WELL-BALANCED FINITE VOLUME SCHEME FOR 1D HEMODYNAMIC SIMULATIONS ∗

We are interested in simulating blood flow in arteries with variable elasticity with a one dimensional model. We present a well-balanced finite volume scheme based on the recent developments in shallow water equations context. We thus get a mass conservative scheme which also preserves equilibria of Q=0. This numerical method is tested on analytical tests.